This study presents a mathematical model for analyzing inventory systems with time-dependent deteriorating items and holding costs vary linearly over time. The model accommodates expected shortages with a backlog rate that evolves. Demand is characterized by two distinct functions: a time-dependent quadratic function during periods without shortages and a time-dependent linear function during periods with shortages. Computational Algorithm is formulated to obtain the minimal total cost, and the convexity of the full cost function is established. Additionally provide a numerical example to demonstrate the application of the model. Furthermore, sensitivity analysis of optimal inventory policies is conducted, and the impact of decision variable variations are graphically represented using MATLAB. This comprehensive framework contributes to a deeper understanding of inventory management under fluctuating demand conditions.